Chapter A.3, Problem 31E

Solution

To explain why the restriction $x\ne 3$ is needed in the reduced form of the expression $\frac{3-\frac{7}{x+2}}{1-\frac{1}{x-3}}$ .

Given:

The reduced form of the given expression is $\frac{\left(3x-1\right)\left(x-3\right)}{\left(x+2\right)\left(x-4\right)}$ .

Concept Used:

• A domain is the set of all possible input values for a function.
• To find the domain of a function, set the function equal to y and solve for x . This will give you the set of all x -values for which the function produces a real y -value.
• If a factor is common in the numerator and denominator, then the corresponding zero of that factor is excluded from the domain of the original expression.

Calculation:

Note that the factor $\left(x-3\right)$ is common in the numerator and denominator of the factored form of original expression, so the corresponding zeros must be excluded from the domain of the original expression.

The zeros corresponding to this common factor is $x=3$ .

Since the original expression is undefined for $x=3$ , the restriction $x\ne 3$ is needed on the reduced form.